Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

Abstract

Eisenbud, Popescu, and Walter [4] have constructed certain special sextic hypersurfaces in P5 as Lagrangian degeneracy loci. We prove that the natural double cover of a generic Eisenbud-Popescu-Walter (EPW) sextic is a deformation of the Hilbert square of a K3-surface (K3)[2] and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1); thus we get an example similar to that (discovered by Beauville and Donagi [2]) of the Fano variety of lines on a cubic 4-fold. Conversely, suppose that X is a numerical (K3)[2], suppose that H is an ample divisor on X of square 2 for Beauville's quadratic form, and suppose that the map X⇢|H|∨ is the composition of the quotient X→Y for an antisymplectic involution on X followed by an immersion Y↪|H|∨; then Y is an EPW sextic, and X→Y is the natural double cover